Trig3.5

Prove that $\cos (\frac{\pi}{11}) \cos (\frac{2\pi}{11}) \cos (\frac{3\pi}{11}) \cos (\frac{4\pi}{11}) \cos (\frac{5\pi}{11})=\frac{1}{32}\,$

Let $\frac{\pi}{11}=A\,$ so that $\pi=11A\,$

Let $C=\cos A \cos 2A \cos 3A \cos 4A \cos 5A\,$

$S=\sin A \sin 2A \sin 3A \sin 4A \sin 5A\,$

$C\cdot S= \frac{1}{2}\sin 2A \cdot \frac{1}{2}\sin 4A \cdot \sin 6A \cdot 8A \cdot \sin 10A\,$

$\frac{1}{32}\sin 2A \cdot \sin 4A \cdot \sin (\pi-5A)\cdot \sin (\pi-3A) \cdot \sin (\pi-A)\,$

$\frac{1}{32}\sin 2A \cdot \sin 4A \cdot \sin 5A \cdot \sin 3A \cdot \sin A\,$

$\frac{1}{32}S\,$

Therefore, $C\cdot S=\frac{1}{32} S\,$

$C=\frac{1}{32}\,$ =RHS